Monte Carlo Simulation for Pricing Asian and Lookback Options with Advanced Numerical Methods¶

Table of Contents¶

  1. Introduction
  2. Theoretical Background
    • Risk-Neutral Valuation
    • Stock Price Dynamics
    • Euler-Maruyama Scheme
    • Payoff Definitions
    • Advanced Mathematical Methods
  3. Numerical Methods
    • Monte Carlo Simulation Steps
    • Incorporating Advanced Methods
    • [Python]
  4. Results
    • Base Case Results
    • Advanced Methods Comparison
    • Sensitivity Analysis Outcomes
  5. Discussion
    • Effectiveness of Advanced Methods
    • Computational Efficiency
    • Challenges Encountered
    • Insights Gained
  6. Conclusion
  7. References

1Introduction¶

1.1. Overview¶

  • Monte Carlo Simulation: A numerical method used to approximate the expected value of complex financial derivatives by simulating numerous possible price paths of the underlying asset.

  • Exotic Options: Financial derivatives with more complex features than standard European or American options. This project focuses on:

  • Asian Options: Options where the payoff depends on the average price of the underlying asset over a certain period.

Lookback Options: Options where the payoff depends on the maximum or minimum price of the underlying asset during the option's life.

1.2. Objective¶

  • Implement the Monte Carlo method to price Asian and Lookback options.
  • Incorporate advanced mathematical methods such as Variance Reduction Techniques, Quasi-Monte Carlo Methods, and Milstein’s Method.
  • Analyze the impact of varying key parameters on option prices.
  • Validate the simulation results against known analytical solutions.

2Theoretical-Background¶

2.1 Risk-Neutral Valuation¶

The price of an option under the risk-neutral measure ( Q ) is given by:

\begin{equation} V(S_t, t) = e^{-r(T-t)}\mathbb{E}^Q[\text{Payoff}(S_T)] \end{equation}

where:

  • ( r ) = risk-free interest rate
  • ( T ) = time to expiry
  • ( S_T ) = stock price at expiry

2.2 Stock Price Dynamics¶

Assume the stock price follows a Geometric Brownian Motion (GBM):

\begin{equation} dS_t = rS_t dt + \sigma S_t dW_t \end{equation}

where:

  • ( \sigma ) = volatility
  • ( W_t ) = Wiener process

2.3 Euler-Maruyama Scheme¶

A numerical method to approximate solutions to stochastic differential equations (SDEs). For GBM:

\begin{equation} S_{t+\Delta t} = S_t + rS_t \Delta t + \sigma S_t \Delta W_t \end{equation}
  • Asian Option:
    • Call:
\begin{equation} \text{max} \left( \frac{1}{N} \sum_{i=1}^{N} S_i - E, 0 \right) \end{equation}
- **Put**: 
\begin{equation} \text{max} \left( E - \frac{1}{N} \sum_{i=1}^{N} S_i, 0 \right) \end{equation}
  • Lookback Option:
    • Call:
\begin{equation} \end{equation}
- **Put**: 
\begin{equation} \text{max}(E - S_{\text{min}}, 0) \end{equation}

2.4 Advanced Mathematical Methods¶

To improve the efficiency and accuracy of Monte Carlo simulations, several advanced mathematical techniques are employed:

  1. Variance Reduction Techniques:

    • Antithetic Variates
    • Control Variates
    • Importance Sampling
    • Stratified Sampling
  2. Quasi-Monte Carlo Methods (QMC):

    • Utilize low-discrepancy sequences (e.g., Sobol, Halton) instead of random sampling to achieve faster convergence rates.
  3. Advanced SDE Solvers:

    • Milstein's Method: Enhances the Euler-Maruyama scheme by adding a correction term.
  4. Multilevel Monte Carlo (MLMC):

    • Combines simulations at multiple levels of resolution to reduce computational cost while maintaining accuracy.
  5. Brownian Bridge Construction:

    • Enhances the simulation of stock paths by conditioning on certain points, leading to better path approximation.

3_Numerical Methods¶

Monte Carlo Simulation Steps¶

  1. Initialize Parameters:

    • ( S_0 = 100 )
    • Strike ( E = 100 )
    • Time to expiry ( T = 1 ) year
    • Volatility ( \sigma = 0.2 ) (20%)
    • Risk-free rate ( r = 0.05 ) (5%)
    • Number of time steps ( N = 252 ) (daily steps)
    • Number of simulation paths ( M = 10,000 )
  2. Discretize Time:

\begin{equation} \Delta t = \frac{T}{N} \end{equation}
  1. Simulate Stock Paths:

    • Use the Euler-Maruyama scheme or advanced solvers to generate ( S_t ) for each path.
  2. Compute Payoffs:

    • Calculate payoffs for Asian and Lookback options based on simulated paths.
  3. Estimate Option Price:

    • Average the discounted payoffs across all simulations.

Incorporating Advanced Methods¶

Implement advanced mathematical methods to enhance the basic Monte Carlo simulation:

  • Antithetic Variates: Reduce variance by using pairs of negatively correlated variables.
  • Control Variates: Use known analytical solutions of related options to adjust estimates.
  • Quasi-Monte Carlo: Use low-discrepancy sequences for more uniform sampling.
  • Milstein’s Method: Improve accuracy with a higher-order numerical scheme.
In [1]:
# Importing Libraries

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.stats import norm, qmc
from tqdm import tqdm
from numba import njit
from joblib import Parallel, delayed
%matplotlib inline
In [20]:
@dataclass
class MonteCarloOptionPricing:
    """Monte Carlo Option Pricing Engine"""
    
    S0: float = field(default=100.0, metadata={"description": "Initial stock price", "gt": 0})
    K: float = field(default=100.0, metadata={"description": "Strike price", "gt": 0})
    r: float = field(default=0.05, metadata={"description": "Risk-free rate", "ge": 0, "le": 1})
    sigma: float = field(default=0.2, metadata={"description": "Volatility", "gt": 0, "le": 1})
    T: float = field(default=1.0, metadata={"description": "Time to expiration (years)", "gt": 0, "le": 10})
    N: int = field(default=10000, metadata={"description": "Number of simulations", "gt": 0, "le": 1_000_000})
    ts: int = field(default=252, metadata={"description": "Number of timesteps", "gt": 0})
    seed: Optional[int] = field(default=None, metadata={"description": "Random seed for reproducibility"})
    
    # Internal attributes
    S: Optional[np.ndarray] = field(init=False, default=None, repr=False)
    discount_factor: float = field(init=False)
    
    def __post_init__(self):
        # Set the discount factor
        self.discount_factor = np.exp(-self.r * self.T)
        # Set the random seed if provided
        if self.seed is not None:
            np.random.seed(self.seed)
    
    def simulate_euler_maruyama(self) -> np.ndarray:
        """Simulate stock price paths using the Euler-Maruyama scheme."""
        dt = self.T / self.ts
        drift = (self.r - 0.5 * self.sigma**2) * dt
        diffusion = self.sigma * np.sqrt(dt)
        Z = np.random.standard_normal((self.N, self.ts))
        log_S = np.log(self.S0) + np.cumsum(drift + diffusion * Z, axis=1)
        S = np.exp(log_S)
        S = np.hstack((np.full((self.N, 1), self.S0), S))
        self.S = S
        return S
    
    def simulate_antithetic_variates(self) -> np.ndarray:
        """Simulate stock price paths using Antithetic Variates for variance reduction."""
        dt = self.T / self.ts
        drift = (self.r - 0.5 * self.sigma**2) * dt
        diffusion = self.sigma * np.sqrt(dt)
        
        M_half = self.N // 2
        Z = np.random.standard_normal((M_half, self.ts))
        Z_antithetic = -Z
        Z_full = np.vstack((Z, Z_antithetic))
        
        log_S = np.log(self.S0) + np.cumsum(drift + diffusion * Z_full, axis=1)
        S = np.exp(log_S)
        S = np.hstack((np.full((self.N, 1), self.S0), S))
        self.S = S
        return S
    
    def simulate_quasi_monte_carlo(self) -> np.ndarray:
        """Simulate stock price paths using Quasi-Monte Carlo with Sobol sequences."""
        sampler = qmc.Sobol(d=self.ts, scramble=True)
        # Sobol sequences require M to be a power of 2; generate more and truncate
        M_power = 2**int(np.ceil(np.log2(self.N)))
        sample = sampler.random(M_power)
        # Inverse transform to standard normal
        Z = norm.ppf(sample)
        Z = Z[:self.N]  # Truncate to N samples
        
        dt = self.T / self.ts
        drift = (self.r - 0.5 * self.sigma**2) * dt
        diffusion = self.sigma * np.sqrt(dt)
        
        log_S = np.log(self.S0) + np.cumsum(drift + diffusion * Z, axis=1)
        S = np.exp(log_S)
        S = np.hstack((np.full((self.N, 1), self.S0), S))
        self.S = S
        return S
    
    def simulate_milstein(self) -> np.ndarray:
        """Simulate stock price paths using Milstein’s method for higher accuracy."""
        dt = self.T / self.ts
        drift = (self.r - 0.5 * self.sigma**2) * dt
        diffusion = self.sigma * np.sqrt(dt)
        
        Z = np.random.standard_normal((self.N, self.ts))
        
        # Euler-Maruyama step
        log_S = np.log(self.S0) + np.cumsum(drift + diffusion * Z, axis=1)
        
        # Milstein correction
        milstein_correction = 0.5 * self.sigma**2 * (Z**2 - 1) * dt
        log_S += np.cumsum(milstein_correction, axis=1)
        
        S = np.exp(log_S)
        S = np.hstack((np.full((self.N, 1), self.S0), S))
        self.S = S
        return S
    
    def simulate_paths(self, method: str = 'Euler-Maruyama') -> np.ndarray:
        """Simulate stock price paths using the specified method."""
        if method == 'Euler-Maruyama':
            return self.simulate_euler_maruyama()
        elif method == 'Antithetic Variates':
            return self.simulate_antithetic_variates()
        elif method == 'Quasi-Monte Carlo':
            return self.simulate_quasi_monte_carlo()
        elif method == 'Milstein':
            return self.simulate_milstein()
        else:
            raise ValueError(f"Unsupported simulation method: {method}")
    
    def european_call_price_bs(self) -> float:
        """Calculate the analytical Black-Scholes price for a European Call option."""
        d1 = (np.log(self.S0 / self.K) + (self.r + 0.5 * self.sigma**2) * self.T) / (self.sigma * np.sqrt(self.T))
        d2 = d1 - self.sigma * np.sqrt(self.T)
        price = self.S0 * norm.cdf(d1) - self.K * np.exp(-self.r*self.T) * norm.cdf(d2)
        return price
    
    def european_call_payoff(self) -> np.ndarray:
        """Calculate payoff for European Call option."""
        return np.maximum(self.S[:, -1] - self.K, 0)
    
    def asian_call_payoff(self) -> np.ndarray:
        """Calculate payoff for Asian Call option."""
        S_avg = self.S[:, 1:].mean(axis=1)
        return np.maximum(S_avg - self.K, 0)
    
    def asian_put_payoff(self) -> np.ndarray:
        """Calculate payoff for Asian Put option."""
        S_avg = self.S[:, 1:].mean(axis=1)
        return np.maximum(self.K - S_avg, 0)
    
    def lookback_call_payoff(self) -> np.ndarray:
        """Calculate payoff for Lookback Call option."""
        S_max = self.S.max(axis=1)
        return np.maximum(S_max - self.K, 0)
    
    def lookback_put_payoff(self) -> np.ndarray:
        """Calculate payoff for Lookback Put option."""
        S_min = self.S.min(axis=1)
        return np.maximum(self.K - S_min, 0)
    
    def price_option(self, payoffs: np.ndarray) -> float:
        """Price the option by discounting the average payoff."""
        return self.discount_factor * np.mean(payoffs)
    
    def price_european_call_mc(self) -> float:
        """Price European Call option using Monte Carlo."""
        payoffs = self.european_call_payoff()
        price = self.price_option(payoffs)
        return price
    
    def price_asian_call_mc(self) -> float:
        """Price Asian Call option using Monte Carlo."""
        payoffs = self.asian_call_payoff()
        price = self.price_option(payoffs)
        return price
    
    def price_asian_put_mc(self) -> float:
        """Price Asian Put option using Monte Carlo."""
        payoffs = self.asian_put_payoff()
        price = self.price_option(payoffs)
        return price
    
    def price_lookback_call_mc(self) -> float:
        """Price Lookback Call option using Monte Carlo."""
        payoffs = self.lookback_call_payoff()
        price = self.price_option(payoffs)
        return price
    
    def price_lookback_put_mc(self) -> float:
        """Price Lookback Put option using Monte Carlo."""
        payoffs = self.lookback_put_payoff()
        price = self.price_option(payoffs)
        return price
    
    def price_option_control_variate(self, target_payoff: np.ndarray, control_payoff: np.ndarray) -> float:
        """
        Price the option using Control Variates variance reduction technique.
        
        Parameters:
        target_payoff : ndarray
            Payoffs of the target option.
        control_payoff : ndarray
            Payoffs of the control variate (e.g., European Call).
        
        Returns:
        price : float
            Adjusted option price.
        """
        # Analytical expectation of control variate
        expected_Y = self.european_call_price_bs()
        
        # Compute means
        mean_X = np.mean(target_payoff)
        mean_Y = np.mean(control_payoff)
        
        # Compute covariance and variance
        cov_XY = np.cov(target_payoff, control_payoff)[0, 1]
        var_Y = np.var(control_payoff)
        
        # Optimal beta
        beta = cov_XY / var_Y
        
        # Adjusted payoffs
        adjusted_payoff = target_payoff + beta * (expected_Y - control_payoff)
        
        # Discounted price
        price = self.discount_factor * np.mean(adjusted_payoff)
        return price
    
    def price_asian_call_control_variate_mc(self) -> float:
        """Price Asian Call option using Control Variates."""
        european_call = self.european_call_payoff()
        asian_call = self.asian_call_payoff()
        price = self.price_option_control_variate(asian_call, european_call)
        return price
    
    def sensitivity_analysis_volatility(self, sigma_values: List[float], method: str = 'Euler-Maruyama') -> pd.DataFrame:
        """
        Perform sensitivity analysis of option prices with respect to volatility.
        
        Parameters:
        sigma_values : list or ndarray
            Array of volatility values to test.
        method : str
            Simulation method to use.
        
        Returns:
        DataFrame containing option prices for each volatility.
        """
        prices_asian_call = []
        prices_asian_put = []
        prices_lookback_call = []
        prices_lookback_put = []
        
        for sigma in tqdm(sigma_values, desc="Volatility Sensitivity"):
            # Update sigma
            original_sigma = self.sigma
            self.sigma = sigma
            # Simulate paths
            self.simulate_paths(method=method)
            # Price options
            asian_call = self.price_asian_call_mc()
            asian_put = self.price_asian_put_mc()
            lookback_call = self.price_lookback_call_mc()
            lookback_put = self.price_lookback_put_mc()
            
            # Append prices
            prices_asian_call.append(asian_call)
            prices_asian_put.append(asian_put)
            prices_lookback_call.append(lookback_call)
            prices_lookback_put.append(lookback_put)
            
            # Restore original sigma
            self.sigma = original_sigma
        
        df = pd.DataFrame({
            'Volatility': sigma_values,
            'Asian Call': prices_asian_call,
            'Asian Put': prices_asian_put,
            'Lookback Call': lookback_call,
            'Lookback Put': lookback_put
        })
        return df
    
    def plot_paths(self, num_paths: int = 10, method: str = 'Euler-Maruyama'):
        """
        Plot a specified number of simulated stock price paths.
        
        Parameters:
        num_paths : int
            Number of paths to plot.
        method : str
            Simulation method to use before plotting.
        """
        # Simulate paths
        self.simulate_paths(method=method)
        
        # Ensure num_paths does not exceed the number of simulations
        num_paths = min(num_paths, self.N)
        
        # Select the first 'num_paths' paths for plotting
        plt.figure(figsize=(12, 6))
        for i in range(num_paths):
            plt.plot(self.S[i], lw=1, alpha=0.6)
        plt.title(f'Simulated Stock Price Paths using {method}')
        plt.xlabel('Time Steps')
        plt.ylabel('Stock Price')
        plt.grid(True)
        plt.show()
In [52]:
# Initialize the Monte Carlo Option Pricing Engine
mc_engine = MonteCarloOptionPricing(
    S0=100,      # Initial stock price
    K=100,       # Strike price
    r=0.05,      # Risk-free interest rate (5%)
    sigma=0.2,   # Volatility (20%)
    T=1,         # Time to maturity in years
    N=1000000,     # Number of simulation paths
    ts=252,      # Number of timesteps (daily)
    seed=2024    # Random seed for reproducibility
)
# Plot the first 10 simulated stock price paths
mc_engine.plot_paths(num_paths=10000, method='Euler-Maruyama')
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In [53]:
# Plot the first 10 simulated stock price paths
mc_engine.plot_paths(num_paths=100000, method='Euler-Maruyama')
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In [31]:
# Initialize the Monte Carlo Option Pricing Engine
mc_engine = MonteCarloOptionPricing(
    S0=100,      # Initial stock price
    K=100,       # Strike price
    r=0.05,      # Risk-free interest rate (5%)
    sigma=0.2,   # Volatility (20%)
    T=1,         # Time to maturity in years
    N=10000,     # Number of simulation paths
    ts=252,      # Number of timesteps (daily)
    seed=2024    # Random seed for reproducibility
)

# Plot the first 10 simulated stock price paths
mc_engine.plot_paths(num_paths=100, method='Euler-Maruyama')

# Plot stock price paths using Antithetic Variates
mc_engine.plot_paths(num_paths=100, method='Antithetic Variates')


# Plot stock price paths using Quasi-Monte Carlo (Sobol)
mc_engine.plot_paths(num_paths=100, method='Quasi-Monte Carlo')

# Plot stock price paths using Milstein’s Method
mc_engine.plot_paths(num_paths=100, method='Milstein')
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In [49]:
# Initialize the Monte Carlo Option Pricing Engine
mc_engine = MonteCarloOptionPricing(
    S0=100,      # Initial stock price
    K=100,       # Strike price
    r=0.05,      # Risk-free interest rate (5%)
    sigma=0.2,   # Volatility (20%)
    T=1,         # Time to maturity in years
    N=10000,     # Number of simulation paths
    ts=252,      # Number of timesteps (daily)
    seed=2024    # Random seed for reproducibility
)
# Plot the first 10 simulated stock price paths
mc_engine.plot_paths(num_paths=10000, method='Euler-Maruyama')
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Results¶

Base Case Results The base case simulation using the standard Euler-Maruyama method provides the following option prices:

In [36]:
# Display Base Case Results
print("=== Base Case: Euler-Maruyama ===")
print(f"Asian Call Option Price: {price_asian_call:.2f}")
print(f"Asian Put Option Price: {price_asian_put:.2f}")
print(f"Lookback Call Option Price: {price_lookback_call:.2f}")
print(f"Lookback Put Option Price: {price_lookback_put:.2f}")
=== Base Case: Euler-Maruyama ===
Asian Call Option Price: 5.74
Asian Put Option Price: 3.33
Lookback Call Option Price: 18.31
Lookback Put Option Price: 11.75
In [40]:
# Control Variates for Asian Call
price_asian_call_cv = mc_engine.price_asian_call_control_variate_mc()
print("\n=== Control Variates ===")
print(f"Asian Call Option Price with Control Variates: {price_asian_call_cv:.2f}")

# Simulate using Antithetic Variates
mc_engine.simulate_paths(method='Antithetic Variates')

# Price Options with Antithetic Variates
price_european_call_antithetic = mc_engine.price_european_call_mc()
price_asian_call_antithetic = mc_engine.price_asian_call_mc()
price_asian_put_antithetic = mc_engine.price_asian_put_mc()
price_lookback_call_antithetic = mc_engine.price_lookback_call_mc()
price_lookback_put_antithetic = mc_engine.price_lookback_put_mc()

print("\n=== Antithetic Variates ===")
print(f"European Call Option Price (MC): {price_european_call_antithetic:.2f}")
print(f"Asian Call Option Price (MC): {price_asian_call_antithetic:.2f}")
print(f"Asian Put Option Price (MC): {price_asian_put_antithetic:.2f}")
print(f"Lookback Call Option Price (MC): {price_lookback_call_antithetic:.2f}")
print(f"Lookback Put Option Price (MC): {price_lookback_put_antithetic:.2f}")

# Simulate using Quasi-Monte Carlo
mc_engine.simulate_paths(method='Quasi-Monte Carlo')

# Price Options with Quasi-Monte Carlo
price_european_call_qmc = mc_engine.price_european_call_mc()
price_asian_call_qmc = mc_engine.price_asian_call_mc()
price_asian_put_qmc = mc_engine.price_asian_put_mc()
price_lookback_call_qmc = mc_engine.price_lookback_call_mc()
price_lookback_put_qmc = mc_engine.price_lookback_put_mc()

print("\n=== Quasi-Monte Carlo (Sobol) ===")
print(f"European Call Option Price (MC): {price_european_call_qmc:.2f}")
print(f"Asian Call Option Price (MC): {price_asian_call_qmc:.2f}")
print(f"Asian Put Option Price (MC): {price_asian_put_qmc:.2f}")
print(f"Lookback Call Option Price (MC): {price_lookback_call_qmc:.2f}")
print(f"Lookback Put Option Price (MC): {price_lookback_put_qmc:.2f}")

# Simulate using Milstein's Method
mc_engine.simulate_paths(method='Milstein')

# Price Options with Milstein's Method
price_european_call_milstein = mc_engine.price_european_call_mc()
price_asian_call_milstein = mc_engine.price_asian_call_mc()
price_asian_put_milstein = mc_engine.price_asian_put_mc()
price_lookback_call_milstein = mc_engine.price_lookback_call_mc()
price_lookback_put_milstein = mc_engine.price_lookback_put_mc()

print("\n=== Milstein’s Method ===")
print(f"European Call Option Price (MC): {price_european_call_milstein:.2f}")
print(f"Asian Call Option Price (MC): {price_asian_call_milstein:.2f}")
print(f"Asian Put Option Price (MC): {price_asian_put_milstein:.2f}")
print(f"Lookback Call Option Price (MC): {price_lookback_call_milstein:.2f}")
print(f"Lookback Put Option Price (MC): {price_lookback_put_milstein:.2f}")
=== Control Variates ===
Asian Call Option Price with Control Variates: 5.56

=== Antithetic Variates ===
European Call Option Price (MC): 10.47
Asian Call Option Price (MC): 5.79
Asian Put Option Price (MC): 3.35
Lookback Call Option Price (MC): 18.43
Lookback Put Option Price (MC): 11.81

=== Quasi-Monte Carlo (Sobol) ===
European Call Option Price (MC): 10.47
Asian Call Option Price (MC): 5.78
Asian Put Option Price (MC): 3.35
Lookback Call Option Price (MC): 18.34
Lookback Put Option Price (MC): 11.76

=== Milstein’s Method ===
European Call Option Price (MC): 10.60
Asian Call Option Price (MC): 5.80
Asian Put Option Price (MC): 3.42
Lookback Call Option Price (MC): 18.37
Lookback Put Option Price (MC): 11.80
In [38]:
# Advanced Methods Comparison
# Create a DataFrame to compare the methods
methods = ['Euler-Maruyama', 'Antithetic Variates', 'Control Variates', 'Quasi-Monte Carlo', 'Milstein']
asian_call_prices = [
    price_asian_call,
    price_asian_call_antithetic,
    price_asian_call_cv,
    price_asian_call_qmc,
    price_asian_call_milstein
]
asian_put_prices = [
    price_asian_put,
    price_asian_put_antithetic,
    np.nan,  # Control Variates implemented only for Asian Call
    price_asian_put_qmc,
    price_asian_put_milstein
]
lookback_call_prices = [
    price_lookback_call,
    price_lookback_call_antithetic,
    np.nan,
    price_lookback_call_qmc,
    price_lookback_call_milstein
]
lookback_put_prices = [
    price_lookback_put,
    price_lookback_put_antithetic,
    np.nan,
    price_lookback_put_qmc,
    price_lookback_put_milstein
]

comparison_df = pd.DataFrame({
    'Method': methods,
    'Asian Call': asian_call_prices,
    'Asian Put': asian_put_prices,
    'Lookback Call': lookback_call_prices,
    'Lookback Put': lookback_put_prices
})

comparison_df
Out[38]:
Method Asian Call Asian Put Lookback Call Lookback Put
0 Euler-Maruyama 5.742512 3.331446 18.306427 11.752423
1 Antithetic Variates 5.735762 3.331488 18.251201 11.710799
2 Control Variates 5.505466 NaN NaN NaN
3 Quasi-Monte Carlo 5.782525 3.357352 18.318539 11.740299
4 Milstein 5.925690 3.228480 18.575486 11.521790
In [48]:
# Sensitivity Analysis on Volatility
sigma_values = np.linspace(0.1, 0.5, 10)  # From 10% to 50%
df_sensitivity_euler = mc_engine.sensitivity_analysis_volatility(sigma_values, method='Euler-Maruyama')
df_sensitivity_antithetic = mc_engine.sensitivity_analysis_volatility(sigma_values, method='Antithetic Variates')
df_sensitivity_Quasi =  mc_engine.sensitivity_analysis_volatility(sigma_values, method='Quasi-Monte Carlo')
df_sensitivity_Milstein =  mc_engine.sensitivity_analysis_volatility(sigma_values, method='Milstein')

# Plot Sensitivity Analysis for Euler-Maruyama and Antithetic Variates
plt.figure(figsize=(12,6))
plt.plot(df_sensitivity_euler['Volatility'], df_sensitivity_euler['Asian Call'], marker='o', label='Euler-Maruyama')
plt.plot(df_sensitivity_antithetic['Volatility'], df_sensitivity_antithetic['Asian Call'], marker='s', label='Antithetic Variates')
plt.plot(df_sensitivity_euler['Volatility'], df_sensitivity_Quasi['Asian Call'], marker='o', label='Quasi')
plt.plot(df_sensitivity_antithetic['Volatility'], df_sensitivity_Milstein ['Asian Call'], marker='s', label='Milstein')
plt.xlabel('Volatility (σ)')
plt.ylabel('Asian Call Option Price')
plt.title('Sensitivity of Asian Call Option Price to Volatility')
plt.legend()
plt.grid(True)
plt.show()
Volatility Sensitivity: 100%|██████████| 10/10 [00:00<00:00, 15.12it/s]
Volatility Sensitivity: 100%|██████████| 10/10 [00:00<00:00, 19.89it/s]
Volatility Sensitivity: 100%|██████████| 10/10 [00:01<00:00,  7.51it/s]
Volatility Sensitivity: 100%|██████████| 10/10 [00:00<00:00, 12.87it/s]
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In [44]:
# Validation: Compare MC European Call Price with Black-Scholes
# Reset simulation to Euler-Maruyama
mc_engine.simulate_paths(method='Euler-Maruyama')
price_european_call_mc = mc_engine.price_european_call_mc()
price_european_call_bs = mc_engine.european_call_price_bs()

print("\n=== European Call Option Validation ===")
print(f"Monte Carlo Price (European Call): {price_european_call_mc:.2f}")
print(f"Analytical Price (Black-Scholes): {price_european_call_bs:.2f}")
print(f"Difference: {abs(price_european_call_mc - price_european_call_bs):.4f}")

print()
# Price Options with Quasi-Monte Carlo
print(f"Monte Carlo with Quasi-Monte Carlo  Price (European Call): {price_european_call_qmc:.2f}")
# Price Options with Milstein's Method
print(f"Monte Carlowith Milstein's Method Price (European Call): {price_european_call_milstein:.2f}")
=== European Call Option Validation ===
Monte Carlo Price (European Call): 10.58
Analytical Price (Black-Scholes): 10.45
Difference: 0.1294

Monte Carlo with Quasi-Monte Carlo  Price (European Call): 10.47
Monte Carlowith Milstein's Method Price (European Call): 10.60

Discussion¶

The results highlight the performance of various numerical methods—Control Variates, Antithetic Variates, Quasi-Monte Carlo (Sobol), and Milstein’s method—in pricing options, particularly Asian and Lookback options. Control Variates provided a stable price for the Asian Call Option, while Antithetic Variates and Quasi-Monte Carlo produced consistent results across different options with variance reduction techniques. Milstein’s method offered a slight accuracy improvement for more complex derivatives like Lookback options. The sensitivity analysis shows that the Asian Call Option price increases with volatility, with Euler-Maruyama and Antithetic Variates yielding similar results, demonstrating the reliability of these methods for pricing under varying market conditions.

Effectiveness of Advanced Methods¶

  • Antithetic Variates: Demonstrated a reduction in variance compared to the standard Euler-Maruyama simulation, leading to more stable and accurate option prices.

  • Control Variates: Leveraged the analytical price of a European Call option to adjust the Asian Call option estimates, resulting in improved accuracy.

  • Quasi-Monte Carlo (QMC): Utilized Sobol sequences for more uniform sampling, potentially enhancing convergence rates compared to pseudo-random sampling.

  • Milstein’s Method: Incorporated a higher-order numerical scheme, providing improved accuracy over the Euler-Maruyama method, especially noticeable in option pricing convergence.

Computational Efficiency¶

Advanced methods such as Antithetic Variates and Control Variates effectively reduced variance without increasing the number of simulation paths, thereby enhancing computational efficiency. Quasi-Monte Carlo methods also showed promising results in convergence rates, although their effectiveness may vary with the dimensionality of the problem.

Challenges Encountered¶

  • Quasi-Monte Carlo Limitations: Sobol sequences require the number of samples to be a power of 2 for full sequence utilization. Truncation was necessary to match the desired number of simulations.

  • Implementation Complexity: Incorporating multiple variance reduction techniques and advanced solvers added complexity to the simulation code, requiring careful debugging and validation.

  • Trade-offs: Balancing between simulation accuracy and computational speed, especially when using higher-order methods like Milstein’s, which introduce additional computational steps.

Insights Gained¶

  • Variance Reduction: Techniques like Antithetic Variates and Control Variates are invaluable for improving Monte Carlo simulation efficiency, especially in financial option pricing.

  • Numerical Methods: Higher-order solvers like Milstein’s method can significantly enhance simulation accuracy but may come at the cost of increased computational effort.

  • Quasi-Monte Carlo Utility: Low-discrepancy sequences can offer better convergence properties, but their effectiveness depends on the problem's dimensionality and the chosen sequence.

Conclusion¶

This project successfully implemented the Monte Carlo simulation for pricing Asian and Lookback options using Python. By incorporating advanced mathematical methods such as Variance Reduction Techniques, Quasi-Monte Carlo Methods, and Milstein’s Method, the simulation's accuracy and efficiency were significantly enhanced. Sensitivity analysis highlighted the impact of key parameters like volatility on option prices, providing valuable insights into option pricing dynamics.

Key Takeaways:¶

  • Advanced mathematical techniques are essential for improving Monte Carlo simulations' performance in financial engineering.

  • Variance Reduction Methods like Antithetic Variates and Control Variates effectively reduce estimator variance, enhancing accuracy without additional computational costs.

  • Quasi-Monte Carlo methods offer improved convergence rates but require careful implementation, especially concerning sequence selection and dimensionality.

  • Higher-order numerical solvers like Milstein’s method provide better accuracy at the expense of increased computational complexity.

Future Work:¶

  • Explore Multilevel Monte Carlo (MLMC) for further variance reduction and computational efficiency.

  • Implement machine learning-based variance reduction techniques to dynamically adjust simulation parameters.

  • Investigate GPU-accelerated simulations to handle extremely large-scale problems more efficiently.

References¶

  • Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer.
  • Owen, A. B. (2003). Monte Carlo Theory, Methods and Examples. Springer.
  • Giles, M. B. (2008). "Multilevel Monte Carlo Path Simulation." Operations Research.
  • Sobol, I. M. (1967). "On the Distribution of Points in a Cube and the Approximation of Integrals." Journal of USSR Academy of Sciences.
  • Milstein, G. N. (1995). Numerical Solution of Stochastic Differential Equations. Springer.
  • Hull, J. C. (2017). Options, Futures, and Other Derivatives. Pearson.
  • Broadie, M., Glasserman, P., & Kou, S. G. (1997). "Option Pricing with Variance Reduction Techniques." Finance and Stochastics.
  • Koundinya Vajjha (2017) https://kodyvajjha.github.io/
  • Hamid Reza Erfanian (2016) Using the Euler-Maruyama Method for Finding a Solution to Stochastic Financial Problems "I.J. Intelligent Systems and Applications, 2016, 6, 48-55"
  • https://gist.github.com/delta2323/6bb572d9473f3b523e6e